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Frank Plumpton Ramsey

Frank Plumpton Ramsey (22 February 1903 – 19 January 1930) was a British mathematician who, in addition to mathematics, made significant and precocious contributions in philosophy and economics before his death at the age of 26.

Life

He was born on 22 February 1903 in Cambridge where his father Arthur, also a mathematician, was President of Magdalene College. He was the eldest of two brothers and two sisters, and his brother Michael Ramsey later became Archbishop of Canterbury, (the only one of the four siblings who was to remain Christian). He entered Winchester College in 1915 and later returned to Cambridge to study mathematics at Trinity College. Easy-going, simple and modest, Ramsey had many interests besides his scientific work. Even as a teenager Ramsey showed both his profound abilities and the heterogeneity of the issues that concerned him. His brother Lord Ramsey was well aware of both these facts:

He was interested in almost everything. He was immensely widely read in English literature; he was enjoying classics though he was on the verge of plunging into being a mathematical specialist; he was very interested in politics, and well-informed; he had got a political concern and a sort of left-wing caring-for-the-underdog kind of outlook about politics.

– Michael Ramsey, Quoted in Mellor

Ramsey suffered mildly from depression, and was intellectually interested in psychoanalysis. While writing his dissertation he went to Vienna to be psychoanalysed by Theodor Reik, a disciple of Freud. As one of the justifications for undertaking therapy, he asserted in a letter to his mother that unconscious impulses might even affect the work of a mathematician. In September 1925 he married Lettice Baker, the wedding taking place in a Register Office since Ramsey was, as his wife described him, a ‘militant atheist’. (She subsequently ran a photography practice in Cambridge for many years ("Ramsey and Muspratt").) The marriage produced two daughters. Despite his atheism, Ramsey was quite tolerant towards his brother when the latter decided to become a priest in the Church of England.[1]

Academic life

He returned to England in 1924, and with John Maynard Keynes's support he became a fellow of King's College, Cambridge, being the second person ever to be elected without having previously studied at King's College.

In 1926 he became a university lecturer in mathematics and later a Director of Studies in Mathematics at King's College. When Wittgenstein submitted the Tractatus Logico-Philosophicus as his doctoral thesis at Cambridge, Ramsey was Wittgenstein’s supervisor and G.E. Moore and Bertrand Russell were the examiners. Later on the three of them arranged financial aid for Wittgenstein to help him continue his research work.

In 1929 Ramsey and Wittgenstein regularly discussed issues in mathematics and philosophy with Piero Sraffa, an Italian economist who had been brought to Cambridge by Keynes after Sraffa had aroused Benito Mussolini’s ire by publishing an article critical of the Fascist regime in the Manchester Guardian. The contributions of Ramsey to these conversations were acknowledged by both Sraffa and Wittgenstein in their later work.

Ramsey and Wittgenstein

When I.A. Richards and C. K. Ogden, both Fellows of Magdalene, first met Ramsey, he expressed his interest in learning German. According to Richards, he mastered the language "in almost hardly over a week,"[2] although other sources show he took before that one year of German in school. Ramsey was then able, at the age of 19, to make the first draft of the translation of the German text of Wittgenstein’s Tractatus Logico Philosophicus. Ramsey was impressed by Wittgenstein’s work and after graduating as a Wrangler in the Mathematical Tripos of 1923 he made a journey to Austria to visit Wittgenstein, at that time teaching in a primary school in the small community of Puchberg am Schneeberg. For two weeks Ramsey discussed the difficulties he was facing in understanding the Tractatus. Wittgenstein made some corrections to the English translation in Ramsey’s copy and some annotations and changes to the German text that subsequently appeared in the second edition in 1933.

Work

In 1927 Ramsey published the influential article Facts and Propositions, in which he proposed what is sometimes described as a redundancy theory of truth.

One of the theorems proved by Ramsey in his 1930 paper On a problem of formal logic now bears his name (Ramsey's theorem). While this theorem is the work Ramsey is probably best remembered for, he only proved it in passing, as a minor lemma along the way to his true goal in the paper, solving a special case of the decision problem for first-order logic, namely the decidability of (what is now called) Bernays-Schönfinkel-Ramsey class of first-order logic, as well as a characterization of the spectrum of sentences in this fragment of logic. Alonzo Church would go on to show that the general case of the decision problem for first-order logic is unsolvable (see Church's theorem). A great amount of later work in mathematics was fruitfully developed out of the ostensibly minor lemma, which turned out to be an important early result in combinatorics, supporting the idea that within some sufficiently large systems, however disordered, there must be some order. So fruitful, in fact, was Ramsey's theorem that today there is an entire branch of mathematics, known as Ramsey theory, which is dedicated to studying similar results.

His philosophical works included Universals (1925), Facts and propositions (1927), Universals of law and of fact (1928), Knowledge (1929), Theories (1929), and General propositions and causality (1929). Wittgenstein mentions him in the introduction to his Philosophical Investigations as an influence (but not as great an influence as Piero Sraffa).

Work in economics

Keynes encouraged Ramsey to work on economics as "From a very early age, about sixteen I think, his precocious mind was intensely interested in economic problems" (Keynes, 1933). Ramsey responded to Keynes's urging by writing three papers in economic theory all of which were of fundamental importance, though it was many years before they received their proper recognition by the community of economists.

Ramsey’s three papers, described below in detail, were on subjective probability and utility (1926), optimal taxation (1927) and optimal one-sector economic growth (1928). The economist Paul Samuelson described them in 1970 as "three great legacies - legacies that were for the most part mere by-products of his major interest in the foundations of mathematics and knowledge."

A mathematical theory of saving

This significant paper was published in The Economic Journal, and involved "a strategically beautiful application of the calculus of variations" (Paul Samuelson) in order to determine the optimal amount an economy should invest (save) rather than consume so as to maximize future utility, or in Ramsey’s words "how much of its income should a nation save?" (Ramsey, 1928).

Keynes described the article as "one of the most remarkable contributions to mathematical economics ever made, both in respect of the intrinsic importance and difficulty of its subject, the power and elegance of the technical methods employed, and the clear purity of illumination with which the writer's mind is felt by the reader to play about its subject. The article is terribly difficult reading for an economist, but it is not difficult to appreciate how scientific and aesthetic qualities are combined in it together" (Keynes 1933). The Ramsey model is today acknowledged as the starting point for optimal accumulation theory although its importance was not recognized until many years after its first publication.

The main contributions of the model were firstly the initial question Ramsey posed on how much savings should be and secondly the method of analysis, the intertemporal maximization (optimization) of collective or individual utility by applying techniques of dynamic optimization. Tjalling C. Koopmans and David Cass modified the Ramsey model incorporating the dynamic features of population growth at a steady rate and of Harrod-neutral technical progress again at a steady rate, giving birth to a model named the Ramsey-Cass-Koopmans model where the objective now is to maximize household’s utility function.

A contribution to the theory of taxation

In this paper Ramsey's contribution to economic theory was the elegant concept of Ramsey pricing. This is applicable in situations where a (regulated) monopolist wants to maximize consumer surplus whilst at the same time ensuring that its costs are adequately covered. This is achieved by setting the price such that the markup over marginal cost is inversely proportional to the price elasticity of demand for that good. Like its predecessor this paper was published in The Economic Journal in 1927. Ramsey poses the question that is to be solved at the beginning of the article: "a given revenue is to be raised by proportionate taxes on some or all uses of income, the taxes on different uses being possibly at different rates; how much should these rates be adjusted in order that that the decrement of utility may be a minimum?" (Ramsey 1927). The problem was suggested to him by the economist Arthur Pigou and the paper was Ramsey’s answer to the problem.

Truth and probability

Keynes in his Treatise on Probability (1921) argued against the subjective approach in epistemic probabilities. For Keynes, subjectivity of probabilities doesn’t matter as much, as for him there is an objective relationship between knowledge and probabilities, as knowledge is disembodied and not personal.

Ramsey in his article disagrees with Keynes’s approach as for him there is a difference between the notions of probability in physics and in logic. For Ramsey probability is not related to a disembodied body of knowledge but is related to the knowledge that each individual possesses alone. Thus personal beliefs that are formulated by this individual knowledge govern probabilities leading to the notion of subjective probability. Consequently, subjective probabilities can be inferred by observing actions that reflect individuals' personal beliefs. Ramsey argued that the degree of probability that an individual attaches to a particular outcome can be measured by finding what odds the individual would accept when betting on that outcome.

Ramsey suggested a way of deriving a consistent theory of choice under uncertainty that could isolate beliefs from preferences while still maintaining subjective probabilities.

Despite the fact that Ramsey’s work on probabilities was of great importance again no one paid any attention to it until the publication of Theory of Games and Economic Behavior of John von Neumann and Oskar Morgenstern in 1944 (1947 2nd ed.)

Early death

Suffering from chronic liver problems, Ramsey contracted jaundice after an abdominal operation and died on 19 January 1930 at Guy's Hospital in London at the age of 26.

The Decision Analysis Society annually awards the Frank P. Ramsey Medal to recognise substantial contributions to decision theory and its application to important classes of real decision problems. He is buried at the Parish of the Ascension Burial Ground in Cambridge.

1. ^ "He was certainly sorry that I went on being religious; he was sorry that I decided to become a priest in the Church of England; sorry indeed, but quite tolerant." Quoted in Mellor, "Ramsey", p. 255
2. ^ Quoted in Mellor, "Ramsey", p. 245,